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Micromechanical study of elastic moduli of loose granular materials

N.P. Kruyta & I. Agnolinb & S. Ludinga & L. Rothenburgc

aDepartment of Mechanical Engineering, University of Twente, Enschede, The NetherlandsbDeformation and Rheology Department, GeoForschungsZentrum Potsdam, Potsdam, Germany

cDepartment of Civil Engineering, University of Waterloo, Waterloo, Canada

Journal of the Mechanics and Physics of Solids (accepted for publication)

Abstract

In micromechanics of the elastic behaviour of granular materials, the macro-scale continuum

elastic moduli are expressed in terms of micro-scale parameters, such as coordination number (the

average number of contacts per particle) and interparticle contact stiffnesses in normal and tangen-

tial directions. It is well-known that mean-field theory gives inaccurate micromechanical predictions

of the elastic moduli, especially for loose systems with low coordination number.

Improved predictions of the moduli are obtained here for loose two-dimensional, isotropic assem-

blies. This is achieved by determining approximate displacement and rotation fields from the force

and moment equilibrium conditions for small sub-assemblies of various sizes. It is assumed that the

outer particles of these sub-assemblies move according to the mean field. From the particle displace-

ment and rotation fields thus obtained, approximate elastic moduli are determined. The resulting

predictions are compared with the true moduli, as determined from Discrete Element Method simu-

lations for low coordination numbers and for various values of the tangential stiffness (at fixed value

of the normal stiffness). Using this approach, accurate predictions of the moduli are obtained, espe-

cially when larger sub-assemblies are considered.

As a step towards an analytical formulation of the present approach, it is investigated whether it is

possible to replace the local contact stiffness matrices by a suitable average stiffness matrix. It is

found that this generally leads to a deterioration of the accuracy of the predictions.

Many micromechanical studies predict that the macroscopic bulk modulus is hardly influenced by

the value of the tangential stiffness. It is shown here from Discrete Element Method simulations of

hydrostatic compression that for loose systems, the bulk modulus strongly depends on the stiffness

ratio for small stiffness ratios.


2

1. Introduction

Knowledge of the mechanical behaviour of granular materials is important in many industrial,

geotechnical and geophysical applications. This behaviour is described by the constitutive relation,

which usually is based on a continuum-mechanical approach, and involves heuristic assumptions.

An alternative is the micromechanical approach (or multi-scale approach), in which a granular

material is modelled as an assembly of interacting particles. The challenge is then to predict the

macro-level, continuum behaviour in terms of micro-level characteristics of particles and interparti-

cle contacts. The micromechanical approach therefore incorporates the discrete nature of granular

materials.

Two special features of granular materials that make their behaviour different from many other

materials are that (i) the particles have rotational degrees of freedom, besides the standard transla-

tional degrees of freedom; thus extended continuum mechanical theories are appropriate (see for

example Eringen, 1999) and (ii) the particle interaction is non-central. This can be modelled by tan-

gential elasticity and/or Coulomb friction at interparticle contacts.

Here micromechanics of quasi-static deformation of granular materials is studied. The continuum

constitutive relation then involves (increments of) the stress and strain tensors. The focus is on the

elastic behaviour of two-dimensional isotropic systems at low packing densities. True, elastic and

reversible deformation of cohesionless granular materials is found for very small strains, typically

for strains of % for stress levels encountered in engineering applications (Tatsuoka, 1999) or in

dynamic problems. For larger strains, true reversible elastic behaviour is lost, due to frictional

behaviour and due to disruption and creation of contacts between particles (see for example, Calvetti

et al., 2002; Rothenburg & Kruyt, 2004).

An overview of experimental results and methods on the elastic behaviour of granular materials

is presented by Tatsuoka (1999). Measurements of elastic moduli, using static methods, are pre-

sented by Hoque & Tatsuoka (2004). Magnanimo et al. (2008) list various studies wherein the elas-

tic moduli have been determined from dynamic, wave-speed measurements.

In many micromechanical studies of the elastic moduli (Rothenburg, 1980; Christoffersen et al.,

1981; Digby, 1981; Walton, 1987; Bathurst and Rothenburg, 1988a, b; Chang et al., 1990; Rothen-

burg and Bathurst, 1992; Chang and Liao, 1994; Cambou et al., 1995; Luding, 2004), the uniform

strain assumption (or mean field assumption) is adopted, according to which the relative displace-

ment of two particles in contact is determined by the average displacement gradient and the relative

position vector of the particle centres. However, the corresponding predictions of the moduli gener-

ally significantly overestimate the actual moduli, especially for loose systems for which the coordi-

10 3–

N.P. Kruyt & I. Agnolin & S. Luding & L. Rothenburg

3

nation number (the average number of contacts per particle) of the assembly is low (see for example

Kruyt & Rothenburg, 1998; Makse et al., 1999; Rothenburg & Kruyt, 2001; Agnolin & Roux,

2007b; Somfai et al., 2007; Magnanimo et al., 2008). Using minimum potential energy principles, it

can be shown rigorously that the uniform strain assumption gives an upper bound to the moduli

(Kruyt & Rothenburg, 2002, 2004). Another lower bound for the elastic bulk modulus is given by

Agnolin & Roux (2007b).

As noted before, the relative displacements of particles in contact deviate from that according to

the uniform strain assumption. These deviations, or fluctuations, have been studied systematically

from results of Discrete Element Method (DEM for short) simulations (Kruyt & Rothenburg, 1998;

Agnolin & Roux, 2007b; Agnolin & Kruyt, 2008). Theoretical work has been done to predict these

fluctuations and, as a result, obtain more accurate predictions of the elastic moduli. In most studies

(except Rothenburg & Kruyt, 2009) the fluctuations are determined by solving the displacements

and rotations from the equilibrium equations for very small sub-assemblies. This work is either

purely analytical (Jenkins et al., 2005; Agnolin et al., 2006; La Ragione & Jenkins, 2007) or semi-

numerical (Kruyt & Rothenburg, 2002, 2004; Agnolin & Roux, 2008; Agnolin & Kruyt, 2008).

Such approaches work reasonably well for dense assemblies with high coordination number,

while for loose assemblies with low coordination number large deviations are observed between

results of DEM simulations and theoretical predictions (Kruyt & Rothenburg, 2002; Agnolin &

Roux, 2007b, 2008; Agnolin & Kruyt, 2008). The focus of the current study is therefore on the ‘dif-

ficult’ case of loose assemblies with low coordination number. Previous approaches are extended

here by solving displacements and rotations fluctuations from the equilibrium conditions for sub-

assemblies with various sizes.

The outline of this paper is as follows. In Section 2 an overview is given of micromechanics of

granular materials. In Section 3 the adopted approach is described for the determination of the parti-

cle displacement and rotation field from equilibrium conditions for small sub-assemblies of parti-

cles. From these fields, predictions of the moduli are obtained. In Section 4 the DEM simulations, as

proposed by Cundall & Strack (1979), are described. In Section 5 the predictions from the microme-

chanical approach are compared against those from the DEM simulations.

The usual sign convention from continuum mechanics is employed for stress and strain, accord-

ing to which tensile stresses and strains are considered positive. The summation convention is

adopted, implying a summation over repeated subscripts.


4

2. Micromechanics

In the absence of long-range forces, particles only exert forces on one another when they are in

contact. The particles in the assembly are assumed to be semi-rigid, so that the interparticle interac-

tion can be considered as point forces acting at contacts.

Here assemblies of two-dimensional circular particles, i.e. disks, are considered. The radius of

particle is denoted by . The geometry of two particles and in contact is depicted in Figure

1 for the two-dimensional case: and are the normal and tangential vector at the contact point

, respectively. The position vector of the centre of particle is denoted by . The branch vector

is the vector connecting the centres of two particles in contact, i.e. . The orienta-

tion of a contact is defined by the direction of its branch vector (see

also Figure 1).

Statistical properties of the contact geometry of an assembly are primarily described by the two

quantities: (i) the coordination number , i.e. the average number of contacts per particle, which is

a measure of the packing density (Quickenden & Tan, 1974; Kruyt & Rothenburg, 1996) and (ii) the

contact distribution function (Horne, 1965) that is defined such that gives the fraction

of contacts with orientation within the interval . For the two-dimensional isotropic

assemblies considered here, . Note that there is no one-to-one relation between

packing density and coordination number (Agnolin & Roux, 2007a; Magnanimo et al., 2008).

In the absence of body forces, the quasi-static two-dimensional force and moment equilibrium

conditions for particle are

p Rp p q

nipq ti

pq

c p Xip

lipq li

pq Xiq Xi

p–=

qpq nipq qpqcos qpqsin,( )T=

+

+

Particle q

Particle p

Figure 1. Contact geometry of particles and , involving the contact normal and tangen-

tial vectors and , branch vector and contact orientation .

p q

nipq ti

pq lipq qpq

Contact point c

Xip

Xiq

nipqti

pq

lipq

qpq

Γ

E q( ) E q( )Dq

q q q Dq+,( )

E q( ) 1 2π( )⁄=

p


5

(1)

where is the vector from the centre of particle to the point of contact between particles and

, is the force exerted on particle by particle , the summation is over the particles that

are in contact with particle and is the two-dimensional permutation symbol.

Using an averaging technique for the force equilibrium conditions Eq. (1), the micromechanical

expression for the average (increment of the) Cauchy stress tensor is obtained (see for example,

Drescher & de Josselin de Jong, 1972; Kruyt & Rothenburg, 1996)

(2)

where is the total area occupied by the assembly (including voids) and is the set of contacts.

Note that is a ‘proper’ contact quantity: , since and . The

latter identity is Newton’s third law.

The relative displacement at the contact between particles and consists of two terms,

one due to the displacement of the particle centres and one due to the rotation of particles

(3)

where is the displacement of the centre of particle and is its rotation (counted positive in

counter-clockwise direction). Note that .

The link between (increments of) forces at contacts and (increments of) relative displacements is

through the contact constitutive relation. Since very small strains are considered, it is assumed that:

• The contact topology is fixed (“cemented contacts”), once a certain configuration with a certain

coordination number has been chosen. Thus, contact creation and disruption are not taken into

account. Changes in contact orientation are also ignored.

• Interparticle Coulomb friction is ignored in the following, since the number of interparticle con-

tacts where Coulomb friction is fully mobilised (“sliding contacts”) is very small in the initial iso-

tropic state. Alternatively, one could state that the case of an infinite interparticle friction is

considered.

• The contact elastic properties are constant: there is no variation in stiffness at contacts, contrary to

the case with a non-linear elastic contact constitutive relation (such as a Hertzian-type).

Hence a simple linear elastic contact constitutive relation is employed here

(4)

fipq

q∑ 0= eij ri

pqfjpq

q∑ 0=

ripq p p

q fipq p q q

p eij

σij

σij1A--- fi

cljc

c C∈∑=

A C

ficlj

c fiqplj

qp fipqlj

pq= ljqp lj

pq–= fiqp fi

pq–=

Dipq p q

Dipq Ui

q Ωqejirjqp+[ ] Ui

p Ωpejirjpq+[ ]–=

Uip p Ωp

Diqp Di

pq–=

fic Sij

c Djc= Sij

c knnicnj

c kttictj

c+=


6

where is the contact stiffness matrix, and are the (constant) stiffnesses in normal and tan-

gential directions of the (two-sided) springs.

The relations between particle properties and the contact stiffnesses and is studied in the

field of contact mechanics (see for example Johnson, 1985). The ratio is related to the Pois-

son ratio of the particles (Johnson, 1985). In many studies the case is considered, when for

disks interparticle forces are central and particle rotations are all zero. In the case , force

and relative displacement at contacts are co-linear. The influence of the microscopic stiffness

ratio on the macroscopic Poisson ratio has been studied by Bathurst & Rothenburg (1988a, b)

and Rothenburg et al. (1991).

The process of contact creation and disruption is generally not reversible. Only by fixing the con-

tact topology and by ignoring the frictional Coulomb limit, can true reversible elastic behaviour be

studied (Calvetti et al., 2002).

2.1. Minimum potential energy principle

Variational principles for continuous elastic solids are well established (for example, Hill, 1950;

Washizu, 1968). Analogous discrete variational principles for granular materials have been formu-

lated by Kruyt & Rothenburg (2002, 2004) for systems with non-rotating and rotating particles,

respectively. The discrete minimum potential energy principle reported by Kruyt & Rothenburg

(2004) is summarised here, as it will be used subsequently.

At the boundary of the system the displacements are prescribed. Consider two displacement

and rotation fields and , which both satisfy the displacement boundary condi-

tions on . These displacement and rotation fields have associated relative displacement fields

and , according to Eq. (3). The corresponding contact force fields and are

determined from the contact constitutive relation Eq. (4). By definition, the contact force field

satisfies all equilibrium conditions Eq. (1), contrary to the contact force field .

The discrete minimum potential energy principle is

(5)

This energy principle states that, among all displacement and rotation fields , the true

displacement and rotation field , whose contact force field satisfies the equilibrium

conditions Eq. (1), makes the potential energy density an absolute minimum.

Sijc kn kt

kn kt

kt kn⁄

kt 0=

kt kn⁄ 1=

fic Di

c

kt kn⁄

B

Uip Ωp, Ui

*p Ω*p,

B

Dic Di

*c fic fi

*c

fic

fi*c

Π Uip Ωp, ( ) 1

2A------- fi

cDic

c C∈∑≡ 1

2A------- fi

*cDi*c

c C∈∑ Π Ui

*p Ω*p, ( )≡≤

Ui*p Ω*p,

Uip Ωp, fi

c

Π


7

2.2. Predictions of moduli: energy-based vs. stress-based approaches

Consider the case where the displacement of the particles at the boundary conforms to the

prescribed uniform (increment of the) displacement gradient , i.e. . The

(increment of the) strain tensor equals the symmetric part of the displacement gradient tensor

(6)

The energy density of the true displacement and rotation field will now

be expressed in terms of the strain tensor and the fourth-order elastic stiffness tensor , which

relates stress and strain through

(7)

According to the virtual work principle (Kruyt & Rothenburg, 2002, 2004) and the assumed

boundary displacements, we find

(8)

since the stress tensor is given by (Kruyt & Rothenburg, 1996) and

the stress tensor is symmetric due to the moment equilibrium equations. Thus, the energy density,

Eq. (5), of the true displacement and rotation field is given by .

For the case considered of isotropic assemblies, the elastic stiffness tensor is isotropic. It is

then described by the bulk modulus and the shear modulus . These can be determined by con-

sidering the two loading paths given in Eq. (18). For isotropic compression, the stress tensor is given

by , while for pure shearing . For conciseness of notation, this is expressed

as . Here the elastic modulus is either the bulk modulus or the shear modulus,

depending on the loading path. In terms of this elastic modulus , the energy density Eq. (5) of the

true displacement and rotation field is given by .

By evaluating the energy density of a displacement and rotation field

whose associated contact force field does not satisfy all equilibrium conditions

Eq. (1), a rigorous upper bound for the modulus is obtained

(9)

This is called the energy-based approach for obtaining a prediction of the modulus , based on the

displacement and rotation field . Note that different displacement and rotation fields

Uiβ β B

aij Ui∂ xj∂⁄≡ Uiβ aijXj

β=

eij

eij12--- aij aji+( )=

Π Uip Ωp, ( ) Ui

p Ωp,

eij Lijkl

σij Lijklekl=

ficDi

c

c C∈∑ fi

βUiβ

β B∈∑ fi

βXjβ

β B∈∑

aij Aσijaij Aσijeij= = = =

σij σij 1 A⁄( ) fiβXj

ββ B∈∑=

Π Uip Ωp, ( ) 1 2⁄( )eijLijklekl=

Lijkl

K G

σij 2Keij= σij 2Geij=

σij 2Meij= M

M

Π Uip Ωp, ( ) Meijeij=

Π Ui*p Ω*p, ( )

Ui*p Ω*p, fi

*c

M

Meijeij Π Uip Ωp, ( ) Π Ui

*p Ω*p, ( )≤=

M

Ui*p Ω*p,


8

yield different bounds for the moduli.

Alternatively, from a displacement and rotation field with associated contact force

field , a stress tensor can be determined according to Eq. (2). From this stress tensor the

modulus can be computed from , since the displacement gradient , and hence the

strain tensor , has been prescribed. This is called the stress-based approach.

The derivation of Eq. (2) is based on the equilibrium conditions Eq. (1). The contact force field

generally does not satisfy all equilibrium conditions. Thus, the stress tensor , obtained by

using Eq. (2) , is (strictly) not a true stress tensor.

For the true displacement and rotation field , the energy-based and the stress-based

approaches yield the same modulus.

3. Local neighbourhood approach

3.1. Basic idea

To obtain a prediction of the elastic modulus, a displacement and rotation field is

required. The basic idea in the proposed approach is to determine these fields from the solution of

the force and moment equilibrium equations of neighbourhoods, or small sub-assemblies. These

equations are solved with prescribed displacements and rotations, according to the imposed dis-

placement gradient, of the outer particles of these sub-assemblies.

For a specified particle , this particle together with its direct neighbours, i.e. those particles with

which it is in contact, form a first-order sub-assembly. For a particle , its second-order sub-assem-

bly consists of the particles in the first-order sub-assembly, together with the neighbours of the outer

particles in the first-order sub-assembly (but excluding particles that already belong to the first-order

sub-assembly). This can easily be generalised to larger sub-assembly orders (or sizes), as illustrated

in Figure 2.

For each particle , its corresponding sub-assembly is determined. With prescribed displace-

ments and rotations of the outer particles at the boundary of the sub-assembly (as indicated in Figure

2), the displacements and rotations of all inner particles are determined from the force and moment

equilibrium equations. The formulation of these equilibrium equations is described in detail in Sec-

tion 3.2. Only the displacements and rotation of particle are retained from this analysis. This pro-

cedure is followed for each particle independently. In the end, a displacement and rotation field

is obtained. From these fields, predictions of the moduli are determined, as described in

Section 2.2.

This approach has already been used by Kruyt & Rothenburg (2002, 2004) using first-order sub-

Ui*p Ω*p,

fi*c σij

*

M σij* 2Meij= aij

eij

fi*c σij

*

σij* 1 A⁄( ) fi

*cljc

c C∈∑=

Uip Ωp,

Ui*p Ω*p,

p

p

p

p

Ui*p Ω*p,


9

First-order sub-assembly; m = 1 Second-order sub-assembly; m = 2

Figure 2. Sketch of sub-assemblies for a specified central particle shown in red. Top left:

first-order, top right: second-order, bottom left: third-order, bottom right:

fourth-order sub-assembly. The outer particles of the sub-assembly, shown in

dark blue, are assigned displacements and rotations that conform to the mean-

field deformation. The displacements and rotations of the inner particles are

determined from the equilibrium equations. Particles shown in black are ignored

in the analysis of the sub-assembly for the red particle. Contacts between particles

are denoted by lines connecting their centres. Only part of the full assembly is

shown.

Third-order sub-assembly; m = 3 Fourth-order sub-assembly; m = 4


10

assemblies. Here it is generalised by considering larger sub-assemblies.

3.2. Solution of equilibrium equations for sub-assemblies

Now it is described in detail how the unknown displacements and rotations of the particles inside

the sub-assembly are obtained from the equilibrium conditions Eq. (1) for these particles. This solu-

tion process consists of three steps:

• definition of fluctuations, relative to the mean field, in particle displacements and rotations;

• definition of the “element matrix”, in FEM terminology, that specifies the interaction between

two particles in contact;

• formulation and solution of the system of linear equilibrium equations, in terms of the fluctua-

tions, for the sub-assembly.

3.2.1. Fluctuations

The displacements, , and rotations, , of particles are divided into contributions due to the

imposed displacement gradient and fluctuations

(10)

where is the average system rotation corresponding to the imposed displacement

gradient , while and are the displacement and rotation fluctuation of particle , respec-

tively. As before is the position vector of the centre of particle . Effectively, Eq. (10) repre-

sents a change of variables from to . While the particle displacements are, on

average, position-dependent, the fluctuations are zero on average over all particles.

In the uniform strain assumption, all displacement and rotation fluctuations are taken equal to

zero, i.e. and . Hence the corresponding relative displacements are given by (see

Eq. (3))

(11)

where is the branch vector at contact as defined in Section 2. The superscript is present in

the quantity , since it depends on the imposed displacement gradient .

3.2.2. Element matrix

A contact between two particles and can be considered as a “finite element” (with two

nodes). The interaction between the particles is specified by the “element matrix” in FEM terminol-

Uip Ωp

aij

Uip aijXj

p uip+= Ωp 1

2--- a21 a12–( ) ωp+=

a21 a12–( ) 2⁄

aij uip ωp p

Xip p

Uip Ωp, ui

p ωp,

uip 0≡ ωp 0≡ Di

c a,

Dic a, eijlj

c=

lic c a

Dic a, aij

p q


11

ogy, see for example Zienkiewicz & Taylor (1987). The element matrix relates the generalised dis-

placements at the contact to the generalised load at the contact.

The generalised contact displacement vector (of length 6) and the generalised contact load

vector (of length 6) of a contact are defined by

(12)

where is the tangential component of the force vector exerted by particle on particle .

The generalised contact load vector and the generalised contact displacement vector

are related by

(13)

The symmetrical 6 by 6 element matrix corresponding to the contact constitutive relation Eq.

(4) is given by

(14)

Here the symmetrical 6 by 6 transformation matrix , from local -variables at the con-

tact to Cartesian variables, is given by

(15)

Wpq

Fpq c pq≡

Wpqu1

p u2p ωp u1

q u2q ωq

T=

Fpqf1pq f2

pq Rp ftpq⋅ f1

qp f2qp Rq ft

qp⋅T

=

ftpq fi

pq q p

Fpq Wpq

Fpq Kpq Wpq⋅–=

Kpq

Kpq τ qpq( )T

kn 0 0 kn– 0 0

0 kt Rpkt 0 kt– Rqkt

0 Rpkt RpRpkt 0 Rpkt– RpRqkt

kn– 0 0 kn 0 0

0 kt– Rpkt– 0 kt ktRq–

0 Rqkt RpRqkt 0 ktRq– RqRqkt

τ qpq( )⋅ ⋅=

τ qpq( ) n t,( )

τ q( )

qcos qsin– 0 0 0 0qsin qcos 0 0 0 0

0 0 1 0 0 00 0 0 qcos qsin– 00 0 0 qsin qcos 00 0 0 0 0 1

=


12

3.2.3. Formulation of equilibrium equations for a sub-assembly

The sub-assembly represents a structure, with kinematic boundary conditions according to the

imposed displacement gradient for the outer particles of the sub-assembly. The connectivity of the

nodes (particles) is determined by the contacts. The interaction between the nodes (particles) is

expressed by the element matrix given in Eq. (14). By “assembling” (see the Zienkiewicz & Taylor,

1987) the global system of equilibrium equations for the small sub-assembly from the element

matrices (contacts), the linear system of equations to be solved for the fluctuations is obtained (for-

mally)

(16)

Here is the vector with all unknown fluctuation displacements and fluctuation rotations within

the sub-assembly for particle ; the length of is . The sum is over all contacts within

the set of contacts of the sub-assembly for particle of order . The matrix (of size 6

by ) accounts for the extraction of the global degrees of freedom within the sub-assembly to

the degrees of freedom corresponding to contact . Note that each row of this matrix contains only a

single element equal to one, while the other elements are all zero. The matrix , the transpose of

, accounts for the opposite transformation, going from local contact quantities to the correspond-

ing global quantity in the vector . The vector (of length 6) contains the displacements and

rotations at contact according to the average displacement gradient

(17)

Here and give the ‘head’ (second) and ‘tail’ (first) particles involved in contact and

gives the first component of the vector (of length 2) and other terms are

defined analogously. As before, gives the position vector of the centre of particle . The posi-

tion vector is conveniently expressed as (where is the branch vector of

contact , see Figure 1) to account for ‘wrap-around’ of position vectors in the periodic systems

studied here. In essence, the right-hand side of Eq. (16) represents the loading of the particles due to

the imposed mean field (see also, Kruyt & Rothenburg, 2002, 2004).

For a specified particle and order of the sub-assembly , the vector is computed from the

equilibrium equations Eq. (16) for the inner particles. Only the fluctuation displacement and the

rotation of particle are retained. This procedure is executed for all particles consecutively.

Note that these computations are uncoupled. In this manner an approximation of the

TcT Kc Tc⋅ ⋅c C p m;( )∈

∑ Vp⋅ TcT Kc⋅ Ua c,⋅( )

c C p m;( )∈∑=

Vp

p Vp N p m;( ) c

C p m;( ) p m Tc

N p m;( )

c

TcT

Tc

Vp Ua c,

c a

Ua c,a XT c( )⋅( )1 a XT c( )⋅( )2

12--- a21 a12–( ) a XH c( )⋅( )1 a XH c( )⋅( )2

12--- a21 a12–( )

T=

H c( ) T c( ) c

a XH c( )⋅( )1 a XH c( )⋅

Xp p

XH c( ) XH c( ) XT c( ) lc+= lc

c

a

p m Vp

uip

ωp p

Ui*p Ω*p,


13

true displacement and rotation field is constructed.

The displacements of particles at the boundary are not changed, since these are prescribed

according to the mean field. Since the average displacement gradient is determined by the dis-

placements at the boundary , as follows from Gauss theorem (see also Kruyt

& Rothenburg, 1996), the displacement and rotation fields are always consistent with

the prescribed displacement gradient .

4. DEM simulations

Discrete Element Method simulations, as proposed by Cundall & Strack (1979), have been per-

formed with two loose assemblies of 50,000 disks from a fairly wide lognormal particle-size distri-

bution, for which the ratio of the standard deviation over the mean equals 0.25. The coordination

numbers of the two assemblies are and . The corresponding packing fractions

, i.e. the area occupied by the particles divided by the area of the assembly (including voids), are

and , respectively. Initially, the periodic box is square with a length of about

200 times the average particle diameter. The assemblies are isotropic, as shown by the contact distri-

bution function (defined in Section 2) in Figure 3. The percentage of “rattlers”, i.e. floating

particles without contacts, is 8% and 2% for and , respectively. Note that these

coordination numbers are those obtained after removal of rattlers.

In the DEM simulations used for creating these two initial assemblies, contact creation and dis-

ruption are taken into account (contrary to the DEM simulations reported subsequently). The assem-

bly with has been obtained by a succession of frictionless and frictional isotropic com-

pressions. The assembly with has been obtained by the isotropic compression of a fric-

tionless gas, i.e. with . In both cases a specified pressure has been employed as boundary con-

dition. Periodic boundaries have been used in all DEM simulations to minimize boundary effects.

Starting from these two initial assemblies, DEM simulations with fixed contact topology have

been performed. Thus, contact creation and disruption are not taken into account. The linear elastic

constitutive relation (4) is used, where frictional limits are ignored. Various values for the stiffness

ratio have been considered, for each of the two assemblies, in order to systematically investi-

gate the influence of , with exactly the same contact geometry.

The assemblies were subjected to the loading paths for compression and shear , where

these displacement gradients are given by

Uip Ωp,

B

aij

aij 1 A⁄( ) Uinj BdB∫=

Ui*p Ω*p,

aij

Γ 3.53= Γ 4.08=

η

η 0.756= η 0.829=

E q( )

Γ 3.53= Γ 4.08=

Γ 3.53=

Γ 4.08=

kt 0≡

kt kn⁄

kt kn⁄

aijK aij

G


14

(18)

where is a magnitude of the imposed displacement gradient. Note that the value is not

important, since the adopted model results in linear governing equations.

In the stress-based approach for estimating the elastic moduli (see Section 5.1), the macroscopic

bulk and shear modulus are determined from

(19)

Alternatively, the macroscopic elastic moduli can be expressed in terms of macroscopic Young’s

modulus and Poisson ratio . In the two-dimensional case considered here, and are related

to the bulk and shear moduli by

(20)

According to the uniform strain assumption Eq. (11), the bulk modulus and the shear modu-

lus are given by (see Kruyt & Rothenburg, 1996, 1998)

Figure 3. Polar plots of the contact distribution function for assemblies with coordina-

tion numbers (left) and (right); in blue: actual value based on

bins with width of 1°, in red: isotropic value .

E q( )

Γ 3.53= Γ 4.08=

E q( ) 1 2π( )⁄=

θθ

Γ = 3.53 Γ = 4.08

aijK amag

1 00 1

= aijG amag

1– 00 1

=

amag amag

K G

σ11 σ22+ 2K e11 e22+( )=

σ11 σ22– 2G e11 e22–( )=

Y ν Y ν

νK G–K G+--------------= Y 4KG

K G+--------------=

Ke

Ge


15

(21)

where is the contact number density, i.e. the number of contacts per unit area, and is the aver-

age over all contacts of the length of the branch vector squared. The evolution of the elastic stiffness

tensor according to the uniform strain assumption in biaxial deformation has been studied by Luding

(2004).

The length scales of a granular assembly at which continuum elasticity theory is applicable have

been studied by Goldenberg & Goldhirsch (2002) and Goldhirsch & Goldenberg (2002). Kuhn &

Bagi (2009) have shown that the average initial elastic modulus is hardly influenced by the system

size, i.e. the ratio between average particle radius and size of the periodic box. It must be noted that

these authors use a different contact constitutive relation that allows for contact disruption and crea-

tion, whereas here contacts are fixed (“cemented contacts”). Somfai et al. (2005) have shown, how-

Ke

kn------ cA

4-----ln

2= Ge

kn------ cA

8----- 1

ktkn-----+ ln

2=

Figure 4. Sketch of near-singular configurations. Left: Singular “chain” of unstable red

and green particles that each have two contacts. Right: Near-singular configura-

tion for the red particle with sub-assembly order . The condition number,

i.e. the ratio between the largest and the smallest eigenvalue of the matrix of the

system of equations in Eq. (16), is 1300. The displacement and rotation field cor-

responding to the smallest eigenvalue of this matrix is indicated by vectors for the

displacements of the particles and by dotted lines for the rotation of the particles,

relative to the horizontal. This eigenmode closely corresponds to rolling at the

contacts.

m 2=

cA ln2


16

ever, that sound propagation characteristics are more sensitive to system size.

Visualisation of the displacement fields from the DEM simulations showed some very large par-

ticle displacements fluctuations. These “outliers”, in statistical investigations of the displacement

and rotation fluctuation fields, are due to singular and near-singular configurations. Instead of elimi-

nating these outliers, based on an arbitrary magnitude criterion, some “filtering” of contacts has

been performed on the assemblies to remove such singular and near-singular configurations. The

first employed criterion is that each particle should have at least two contacts. The second criterion

is that no “chains” of particles (see Figure 4 on the left) with two contacts each are allowed. Corre-

sponding contacts are removed and some particles become “rattlers”. The third criterion is that of

the particles that have three contacts, no more than one of the corresponding particles may have two

contacts. This third criterion rules out configurations as sketched in Figure 4 on the right. Thus, the

contacts of the red particle are removed and the red particle becomes a rattler. Consequently, the top

and right green particles also become rattlers, as they have fewer than two contacts. Rattlers are

ignored in all subsequent analyses. By employing these criteria, the number of contacts decreases by

1.5% for and by 0.3% for . Note that these values for the coordination number

are obtained after removal of “rattlers” and after “filtering” out unstable contacts.

The DEM simulations have been performed such that the maximum resultant force and moment

(i.e. the deviations from equilibrium), measured relative to the average normal force (and the aver-

age particle radius for the moment equilibrium equation) acting on each particle, is less than 0.1%.

Γ 3.53= Γ 4.08=

1 2 3 40

5

10

15

20

25

30

m

Nav

g(m)

Γ=3.53Γ=4.08

Figure 5. Average number of particles inside the sub-assembly whose displace-

ments and rotations are computed, for various sub-assembly orders and coor-

dination numbers . The increase of is approximately quadratic in .

Navg m( )

m

Γ Navg m( ) m


17

5. Results

In this section the results for the predictions of the elastic moduli are given. Firstly, the average

number of particles inside the sub-assemblies, whose displacements and rotations have to be com-

puted, is shown in Figure 5 for various sub-assembly orders . This number increases rapidly

(almost quadratically) with .

5.1. Energy-based vs. stress-based approaches

Predictions of the moduli, based on approximate displacement and rotation fields ,

can be obtained in two ways (see Section 2.2): with the energy-based and the stress-based

approaches. These predictions for the shear modulus are compared for various sub-assembly orders

in Figure 6.

With increasing sub-assembly order , both approaches give a more accurate prediction of the

true shear modulus from the DEM simulation. For the case shown in Figure 6 all predictions lie

above the true value. That this holds for the energy-based predictions follows rigorously from the

minimum potential energy principle of Section 2.1. For the stress-based predictions this can not be

shown. Indeed, for very small stiffness ratios the stress-based prediction may drop below the

true DEM-value. Generally, the stress-based predictions are closer to the true value than the energy-

based predictions, see Figure 6. Therefore, the stress-based approach is adopted in the following.

5.2. Predicted bulk and shear moduli

The predictions for the bulk modulus and the shear modulus are shown in Figure 7 for the

coordination numbers and for various values of the stiffness ratio ,

together with the moduli according to the uniform strain assumption Eq. (21) and with the true mod-

uli from the DEM simulations.

The moduli according to the uniform strain assumption differ strongly from the true moduli. This

is especially the case for the shear modulus for the loosest assembly with , where the

average relative deviation is 140%.

The magnitude of the deviations of the predictions from the true DEM-value decrease with

increasing sub-assembly order . The deviations are larger for the shear modulus than for the

bulk modulus . The deviations decrease with increasing coordination number .

Figure 7 shows that the micromechanical predictions capture the dependence of the bulk modulus

on the stiffness ratio . Quantitatively, the bulk modulus is already well predicted for

m

m

Ui*p Ω*p,

m

m

kt kn⁄

K G

Γ 3.53= Γ 4.08= kt kn⁄

G Γ 3.53=

m G

K Γ

G kt kn⁄


18

with average relative deviations of 8% and 4% for and , respectively.

For , these average relative deviations are 4% and 3%, while for , these deviations are

2% and 2%.

Figure 7 shows that the predictions capture the dependence of the shear modulus on the stiff-

ness ratio . Quantitatively, the shear modulus is already fairly well predicted for with

average relative deviations of 38% and 13% for and , respectively. For

, these average relative deviations are 16% and 6%, while for , these deviations are

2% and 2%.

Note that for the lowest value of the stiffness ratio used, the stress-based prediction of the shear

modulus falls slightly below the true DEM-value for . Thus, these predictions do not form

rigorous upper bounds to the moduli, contrary to the energy-based predictions.

5.3. Influence of local vs. average contact stiffness matrix

The equilibrium conditions Eq. (16) for the displacement and rotation fluctuations of the particles

inside a sub-assembly involve the actual contact stiffness matrices of the particles. For possible sim-

plifications of the current approach and, more importantly, for consideration in analytical

approaches, it is interesting to investigate whether it is possible to replace the sum of the local con-

0 1 2 3 4 5 6 7 80.28

0.3

0.32

0.34

0.36

0.38

0.4

0.42

m

G/k

n

Energy-basedStress-basedDEM

Figure 6. Predictions of the shear modulus for various sub-assembly orders for

energy-based and stress-based approaches; coordination number and

stiffness ratio . The prediction for is the shear modulus

according to the uniform strain assumption, Eq. (21). The blue line gives the

result from the corresponding DEM simulation.

G m

Γ 4.08=

kt kn⁄ 0.5= m 0=

m 2= Γ 3.53= Γ 4.08=

m 3= m 4=

G

kt kn⁄ m 2=

Γ 3.53= Γ 4.08=

m 3= m 4=

m 4=


19

tact stiffness matrices of the particles by representative averages without detrimental effects on the

accuracy of the predictions. The local contact stiffness matrix is present in the element matrix repre-

senting the actual particle stiffness matrices on the left-hand side of Eq. (16) and in the right hand-

side of Eq. (16) representing the resultant forces due to mean-field deformation.

For the matrix on the left-hand side of Eq. (16) this simplification amounts to replacing the sum

of the local contact stiffness matrices by an average matrix. This is achieved by replacing

(where the sum is over the particles that are in contact with particle ) by the average ,

where is the number of contacts of particle . The average stiffness matrix is defined by

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

kt / kn

K /

k n

Bulk modulus; Γ = 3.53

DEMUniform strainm=1m=2m=3m=4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

kt / kn

K /

k n

Bulk modulus; Γ = 4.08


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

kt / kn

G /

k n

Shear modulus; Γ = 4.08


Figure 7. Bulk modulus and shear modulus from DEM simulations and according to

the proposed theory for two coordination numbers and for various stiffness

ratios and sub-assembly orders .

K G

Γ

kt kn⁄ m

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

kt / kn

G /

k n

Shear modulus; Γ = 3.53


Kijpq

q∑q p ΓpKij

Γp p Kij


20

(22)

where the latter identity has been obtained by employing the isotropic contact distribution function

and by replacing the local particle radii and in Eq. (14) by the average par-

ticle radius . Note that the element matrix in Eq. (14) depends only on the contact orientation

.

The similar procedure for the right-hand side vector in Eq. (16) gives a vector

(23)

where is the vector of displacements and rotations according to the mean field, as defined in

Eq. (17). The right-hand side equals zero in the preceding equation, since the resultant force due to

the mean-field deformation is zero on average (but not for individual particles). Thus, it is not

meaningful to replace the local contact stiffness matrix in the right-hand side of Eq. (16) by an aver-

age.

After replacing the local stiffness matrix by the average in the left-hand side of Eq. (16), the

resulting equations become

(24)

By solving these equations for each particle consecutively, a displacement and rotation field

is obtained once again, and hence the moduli can be determined.

A comparison of the results employing the local contact stiffness matrices and employing the

average contact stiffness matrix is given in Figure 8. The predictions of the shear modulus

employing the average contact stiffness matrix deviate much more from the true DEM-value than

Kij E q( )Kij q( ) qd0

2π

∫

kn kt+2

--------------- 0 0kn kt+

2---------------– 0 0

0kn kt+

2--------------- 0 0

kn kt+2

---------------– 0

0 0 R2kt 0 0 R2kt

kn kt+2

---------------– 0 0kn kt+

2--------------- 0 0

0kn kt+

2---------------– 0 0

kn kt+2

--------------- 0

0 0 R2kt 0 0 R2kt

= =

E q( ) 1 2π( )⁄= Rp Rq

R Kpq

qpq

Ri E q( )Kij q( )Uja q( ) qd

0

2π

∫ 0= =

Ua c,

TcT K Tc⋅ ⋅c C p m;( )∈

∑ Vp⋅ TcT Kc⋅ Ua c,⋅( )

c C p m;( )∈∑=

Ui*p Ω*p,

G


21

the predictions employing the local contact stiffness matrix, except for . Furthermore,

increasing the sub-assembly order hardly improves the prediction for . Thus, employing an

average contact stiffness matrix significantly degrades the quality of the prediction, except for

.

5.4. ‘Filtering’ vs. suppression of near-singular modes

In Section 4 it has been described how singular and near-singular modes have been suppressed by

‘filtering’ out some contacts according to the criteria given there. An alternative is to retain all con-

tacts and account for the near-singularity when solving the linear system of equations Eq. (16). This

can be done by employing the Moore-Penrose pseudo-inverse (see for example Golub & van Loan,

1996) for the matrix on the left-hand side of Eq. (16). The results are effectively the same, as the rel-

ative differences in predicted moduli between the two approaches are smaller than 0.1%.

5.5. Moduli in the limit case of zero stiffness ratio

In many micromechanical studies, assemblies of disks with , i.e. with only central forces,

are considered (see for example O’ Hern et al., 2003; Majmudar et al., 2007; Ellenbroek et al.,

2009). In these studies scaling relations are observed for coordination number, pressure and bulk

0 1 2 3 4

0.3

0.32

0.34

0.36

0.38

0.4

0.42

m

G/k

n

average stiffnessactual stiffnessDEM

Figure 8. Comparison of the predictions for the shear modulus using the actual contact

stiffness matrix or the average contact stiffness matrix. Results are shown for var-

ious sub-assembly orders , where corresponds to the uniform strain

result Eq. (21). Case with coordination number and stiffness ratio

.

G

m m 0=

Γ 4.08=

kt kn⁄ 0.5=

m 1=

m m 2≥

m 1=

kt 0=


22

and shear moduli near the ‘isostatic’ coordination number (“jamming point”). This ‘isostatic’ limit

, at which the system is statically determinate, corresponds to in the two-dimensional

case considered here when (Roux, 2000). Note that for cases where , stable packings

are possible for in the two-dimensional case. Expressed in terms of coordination number ,

the scaling relations for bulk and shear modulus are

(25)

where and are the exponents in the scaling relation for the bulk and shear modulus, respec-

tively. The exponents and depend on the nature of the particle interaction (O’ Hern et al.,

2003). Generally, it is found that . For the two-dimensional case of linear springs, O’Hern et

al. (2003) find , while Majmudar et al. (2007) give . Note that a small positive

exponent corresponds to a rapid change in modulus near the isostatic coordination number. Further-

more, Ellenbroek et al. (2009) report different scaling exponents for the bulk modulus, depending

on the way the elastic network is generated.

To investigate whether there are significant differences with the case , DEM simulations

with various small values of the stiffness ratio , at constant coordination number, have been

performed of hydrostatic compression. In hydrostatic compression, tangential forces are zero on

average. One would then expect that the tangential stiffness has only a minor effect on the bulk

modulus . In fact, according to the uniform strain assumption Eq. (21), the tangential stiffness

has no effect on the bulk modulus .

Results for the bulk moduli from the DEM simulations are shown in Figure 9. For ,

below the ‘isostatic’ limit, the value of the bulk modulus for is zero and the bulk modu-

lus scales linearly with stiffness ratio. For , with coordination number just above the iso-

static limit, the limit value of the bulk modulus, , is non-zero, although a rapid change

in bulk modulus is observed for small stiffness ratios: the ratio .

For a denser assembly with coordination number (further away from the isostatic limit;

this case has not been considered before), the ratio . For com-

pleteness, qualitatively similar results for the shear modulus are also shown in Figure 9.

These results show that small variations in tangential stiffness have a large quantitative effect

on macroscopic properties, even on the bulk modulus , for which it could be expected that it does

not depend on . This sensitivity is especially pronounced for systems with coordination numbers

below, or near, the isostatic limit. Thus, caution should be exercised when interpreting results from

DEM simulations or theories that employ only central forces ( ).

Γiso Γiso 4=

kt 0= kt 0>

Γ 3≥ Γ

K Γ Γiso–( )rK∝ G Γ Γiso–( )

rG∝

rK rG

rK rG

rG rK>

rK 0.01= rK 0.1=

kt 0>

kt kn⁄

kt

K kt

K

K Γ 3.53=

kt kn⁄ 0→

Γ 4.08=

K kt kn⁄ 0→( )

K kt kn⁄ 1=( ) K kt kn⁄ 0→( )⁄ 40≅

Γ 4.29=

K kt kn⁄ 1=( ) K kt kn⁄ 0→( )⁄ 2.5≅

G

kt

K

kt

kt 0=


23

It has been suggested in another study (Wyart, 2005) that the ratio of shear modulus over

bulk modulus scales linearly with the ‘distance’ from the isostatic coordination number in the

case with

(26)

Note that the ratio is related to the macroscopic Poisson ratio, see Eq. (20).

The scaling relation Eq. (26) has been confirmed by computer simulations for frictionless assem-

blies with (O’ Hern et al., 2003), as well as for frictional assemblies (Somfai et al., 2007).

Since the stiffness ratio represents another measure of ‘distance’ from an isostatic state, it is

interesting to investigate the influence of the stiffness ratio on the ratio of the moduli for small stiff-

10-8 10-6 10-4 10-2 10010-7

10-6

10-5

10-4

10-3

10-2

10-1

100Bulk modulus; Γ = 3.53

k t/kn

K/k

n

Uniform strainDEM

10-8 10-6 10-4 10-2 10010-8

10-6

10-4

10-2

100Shear modulus; Γ = 3.53

k t/kn

G/k

n

Uniform strainDEM

10-8 10-6 10-4 10-2 10010-3

10-2

10-1

100Shear modulus; Γ = 4.08

k t/kn

G/k

n

Uniform strainDEM

Figure 9. Log-log plots of bulk modulus (top row) and shear modulus (bottom row) vs.

stiffness ratio for (left column) and for (right column).

Results are from DEM simulations. Also shown is the prediction Eq. (21) accord-

ing to the uniform strain assumption. For , and scale as

for small stiffness ratios.

K G

kt kn⁄ Γ 3.53= Γ 4.08=

Γ 3.53= K kn⁄ G kn⁄

kt kn⁄

10-8 10-6 10-4 10-2 10010-2

10-1

100Bulk modulus; Γ = 4.08

k t/kn

K/k

n

Uniform strainDEM

G K⁄

Γiso

kt 0=

G Γ( )K Γ( )----------- Γ Γiso–∝

G K⁄

kt 0=

kt kn⁄


24

ness ratios. This is done for the two values of coordination number and . The

results from the DEM simulations are shown in Figure 10. For small stiffness ratio ,

becomes constant.

6. Discussion

A method has been developed for obtaining micromechanical predictions of the macroscopic

elastic moduli of loose two-dimensional isotropic assemblies. The method is based on predictions of

the particle displacement and rotation fields that are determined from the equilibrium equations for

small sub-assemblies of various sizes. From these fields, predictions of the moduli can be obtained

from either a stress-based or an energy-based approach. It is found that generally the stress-based

approach gives a more accurate prediction. It remains an open question why this is the case.

By increasing the order (or size) of the sub-assemblies, the predictions get closer to the true

moduli determined from DEM simulations. The present predictions are much closer to the true mod-

uli than those according to the uniform strain assumption, which can be off by a factor 2–3 for the

loosest assembly with coordination number . The improvement of the prediction for sub-

assembly order is especially pronounced for the shear modulus of the loosest assembly with

. Quantitatively, the shear modulus is already fairly well predicted for with aver-

age relative deviations of 38% and 13% for and , respectively. For ,

these average relative deviations are 16% and 6%, while for , these deviations are 2% and

2%.

Γ 3.53= Γ 4.08=

kt kn⁄

G kt kn⁄( ) K kt kn⁄( )⁄

10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 10010-2

10-1

100

kt/kn

G/K

Γ=3.53Γ=4.08

Figure 10. Ratio of elastic moduli as a function of stiffness ratio for two values

of coordination number .

G K⁄ kt kn⁄

Γ

m

Γ 3.53=

m 2≥

Γ 3.53= m 2=

Γ 3.53= Γ 4.08= m 3=

m 4=


25

These results give indications of the size of sub-assembly that has to be considered, for a

required level of accuracy. In analytical formulations, additional assumptions are required for geo-

metrical averages, like effective average particle stiffnesses. By employing a semi-numerical

approach here, this source of errors is excluded here.

As a step towards an analytical formulation of the present approach, it has been investigated

whether it is possible to replace the sum of the local contact stiffness matrices by a suitable average

contact stiffness matrix in the solution of the equilibrium conditions for the sub-assemblies. It is

found that this leads to a clear deterioration of the predictions, except for the smallest sub-assembly

order . This deterioration may prove to be an impediment to accurate analytical formulations

of the current approach.

It would be expected that the tangential stiffness has only a small influence on the macro-

scopic bulk modulus . Results from DEM simulations of hydrostatic compression of loose sys-

tems, below and slightly above the isostatic limit, and with various small values of the stiffness ratio

show that strongly depends on near . Thus, caution should be exercised

when interpreting results from DEM simulations or theories that employ only central forces

( ). It is observed from the DEM simulations that the ratio of the elastic moduli is con-

stant for small stiffness ratios.

In compressive loading, tangential forces at contacts are zero on average. Therefore the role of

particle rotations is small. In shearing loading, tangential forces at contacts are not zero on average.

An accurate prediction of the particle rotations is difficult, as they are rather sensitive to the contact

geometry (Agnolin & Kruyt, 2008). This forms one reason why it is more difficult to accurately pre-

dict the shear modulus than the bulk modulus.

A different method for predicting the elastic moduli is the “pair-fluctuation” approach (Jenkins et

al., 2005; Agnolin et al., 2006; La Ragione & Jenkins, 2007). There, equilibrium equations are for-

mulated for a pair of particles that correspond to a single contact. From the approximated equilib-

rium equations, the relative displacement at contacts is obtained analytically in terms of geomet-

rical characteristics of the assembly. The two main differences with the current approach are that: (i)

additional assumptions have to be made to facilitate analytical derivations and (ii) the resulting rela-

tive displacements are not necessarily compatible. The relative displacements are compatible

if displacement and rotation fields exist such that the relative displacements are

given by Eq. (3).

In the present approach complete knowledge of the contact geometry is required. This informa-

tion is generally only available from DEM simulations. It is an open question what statistical meas-

ures are sufficient for the description of the contact geometry. For the isotropic systems considered

m

m 1=

kt

K

kt kn⁄ K kt kn⁄ kt kn⁄ 0=

kt 0= G K⁄

Dic

Dic

Uip Ωp, Di

c


26

here the primary statistical measure is coordination number. Two extreme cases are: (i) dense sys-

tems with coordination number around 6 and (ii) loose systems near the isostatic limit. For dense

systems coordination number is considered a sufficient measure of the contact geometry. For loose

systems just above the isostatic limit no statistical measure may be sufficient: the full knowledge of

the contact geometry is required, since a small change in contact geometry leads to large rearrange-

ment of contact forces. The theoretical “pair-fluctuation” approach (as referred to above) involves,

as a statistical measure of the contact geometry, the conditional contact distribution function, which

gives the probability of finding a contact at an orientation given that another contact with certain ori-

entation is present. The theory of La Ragione & Jenkins (2007) additionally involves the standard

deviation in particle coordination number as a statistical measure of the contact geometry. This

standard deviation is directly connected to the coordination number, as was shown by Magnanimo et

al. (2008), using DEM simulations.

The current approach is based on the solution of particle displacements and rotations from the

force and moment equilibrium equations. A dual approach consists of the solution of contact forces

from compatibility equations for relative displacements (Kruyt & Rothenburg, 1996; Kruyt, 2003).

This approach has been pursued analytically by Trentadue (2001, 2004) and semi-numerically by

Kruyt & Rothenburg (2002) for a sub-assembly order for the case without particle rotations.

Extension to the cases with particle rotations and with larger sub-assembly orders will be pur-

sued, as this will yield rigorous lower bounds for the moduli (besides upper bounds based on the

energy-based approach, see Section 5.1), based on the minimum complementary energy principle.

It is interesting to determine correlation functions for the fluctuations in particle displacements

and rotations from the DEM simulations. Maloney (2006) observed that the correlation function

scales with the system size in his study of the elastic response of dense random packings of disks

with central interaction, i.e. with . Thus, in that study particle rotations are all zero, contrary

to the case studied here.

Future work will deal with the extension of the present approach to the study of different particle

shapes and of anisotropic and three-dimensional systems, as well as with analytical formulations of

the current approach.

Acknowledgment

The reviewers are thanked for their valuable comments and suggestions that have helped to

improve the manuscript.

m 1=

m

kt 0=


27

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